Subgroups — C₃×Dic₂₁×C₁₂₇ #44544764

Order 32004.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	Dic₂₁×C₁₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×Dic₇×C₁₂₇	—	3	1	no	maximal
4	C₃²×C₇×C₁₂₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇×C₁₂₇	—	2	1	no	—
6	C₂×C₃×C₇×C₁₂₇	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic
8	C₃×Dic₃×C₁₂₇	—	7	1	no	maximal, Hall
9	Dic₇×C₁₂₇	—	3	1	no	Hall
10	C₃×C₇×C₁₂₇	—	2	1	no	—
11	C₃×C₇×C₁₂₇	Dic₃	1	1	yes	characteristic, complemented
12	C₃×C₇×C₁₂₇	C₃×C₄	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₁₂₇	D₇	1	1	yes	characteristic
14	C₂×C₇×C₁₂₇	C₃×D₃	1	1	yes	characteristic
15	C₃×C₄×C₁₂₇	—	21	1	no	—
16	Dic₃×C₁₂₇	—	7	1	no	—
17	C₃²×C₁₂₇	Dic₇	1	1	yes	characteristic, complemented, Hall
18	C₇×C₁₂₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₁₂₇	—	2	1	no	—
20	C₂×C₃×C₁₂₇	C₃×D₇	1	1	yes	characteristic
21	C₂×C₃×C₁₂₇	D₂₁	1	1	yes	characteristic
22	C₄×C₁₂₇	—	21	1	no	Hall
23	C₃×C₁₂₇	—	2	1	no	—
24	C₃×C₁₂₇	C₃×Dic₇	1	1	yes	characteristic, complemented
25	C₃×C₁₂₇	Dic₂₁	1	1	yes	characteristic, direct product factor
26	C₂×C₁₂₇	C₃×D₂₁	1	1	yes	characteristic
27	C₃×Dic₂₁	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₁₂₇	C₃×Dic₂₁	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₃²×C₇	C₂×C₁₂₇	1	1	yes	characteristic
30	Dic₂₁	C₃×C₁₂₇	1	1	yes	characteristic, direct product factor
31	C₃×Dic₇	—	3	1	no	—
32	C₃²×C₇	C₄×C₁₂₇	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇	—	2	1	no	—
34	C₂×C₃×C₇	D₃×C₁₂₇	1	1	yes	characteristic
35	C₂×C₃×C₇	C₂×C₃×C₁₂₇	1	1	yes	characteristic
36	C₃×Dic₃	—	7	1	no	Hall
37	Dic₇	—	3	1	no	Hall
38	C₃×C₇	—	2	1	no	—
39	C₃×C₇	Dic₃×C₁₂₇	1	1	yes	characteristic, complemented
40	C₃×C₇	C₃×C₄×C₁₂₇	1	1	yes	characteristic, complemented
41	C₂×C₃²	D₇×C₁₂₇	1	1	yes	characteristic
42	C₂×C₇	C₃×D₃×C₁₂₇	1	1	yes	characteristic
43	C₃×C₄	—	21	1	no	—
44	Dic₃	—	7	1	no	—
45	C₃²	Dic₇×C₁₂₇	1	1	yes	characteristic, complemented, Sylow
46	C₇	C₃×Dic₃×C₁₂₇	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₃×D₇×C₁₂₇	1	1	yes	characteristic
49	C₂×C₃	D₂₁×C₁₂₇	1	1	yes	characteristic
50	C₄	—	21	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×Dic₇×C₁₂₇	1	1	yes	characteristic, complemented
53	C₃	Dic₂₁×C₁₂₇	1	1	yes	characteristic, direct product factor
54	C₂	C₃×D₂₁×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₂₇	D₂₁
commutator, hypocenter	C₃×C₇	C₃×C₄×C₁₂₇
Frattini	C₂	C₃×D₂₁×C₁₂₇
Fitting, socle, prime radical	C₂×C₃²×C₇×C₁₂₇	C₂
cosocle	C₂×C₃×C₇	C₂×C₃×C₁₂₇
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃×C₁₂₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Dic₂₁×C₁₂₇	C₃	direct (N × H)
4. C₃²×C₇×C₁₂₇	C₄	semidirect
11. C₃×C₇×C₁₂₇	Dic₃	semidirect
12. C₃×C₇×C₁₂₇	C₃×C₄	semidirect
17. C₃²×C₁₂₇	Dic₇	semidirect
18. C₇×C₁₂₇	C₃×Dic₃	semidirect
24. C₃×C₁₂₇	C₃×Dic₇	semidirect
25. C₃×C₁₂₇	Dic₂₁	direct (N × H)
27. C₃×Dic₂₁	C₁₂₇	direct (N × H)
28. C₁₂₇	C₃×Dic₂₁	direct (N × H)
30. Dic₂₁	C₃×C₁₂₇	direct (N × H)
32. C₃²×C₇	C₄×C₁₂₇	semidirect
39. C₃×C₇	Dic₃×C₁₂₇	semidirect
40. C₃×C₇	C₃×C₄×C₁₂₇	semidirect
45. C₃²	Dic₇×C₁₂₇	semidirect
46. C₇	C₃×Dic₃×C₁₂₇	semidirect
52. C₃	C₃×Dic₇×C₁₂₇	semidirect
53. C₃	Dic₂₁×C₁₂₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₇×C₁₂₇	2
Dic₂₁×C₁₂₇	3
C₃×Dic₇×C₁₂₇	3
C₃×Dic₃×C₁₂₇	7
C₃×Dic₂₁	127

Subgroups — C3×Dic21×C127 #44544764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₂₁×C₁₂₇ #44544764